The Younger Dryas cold period, which interrupted the transition from the last ice age to modern conditions in Greenland, is one of the most dramatic incidents of abrupt climate change reconstructed from paleoclimate proxy records. Changes in the Atlantic Ocean overturning circulation in response to freshwater fluxes from melting ice are frequently invoked to explain this and other past climate changes. Here we propose an alternative mechanism in which the receding glacial ice sheets cause the atmospheric circulation to enter a regime with greater net precipitation in the North Atlantic region. This leads to a significant reduction in ocean overturning circulation, causing an increase in sea ice extent and hence colder temperatures. Positive feedbacks associated with sea ice amplify the cooling. We support the proposed mechanism with the results of a state-of-the-art global climate model. Our results suggest that the atmospheric precipitation response to receding glacial ice sheets could have contributed to the Younger Dryas cooling, as well as to other past climate changes involving the ocean overturning circulation.